Answer to Question #298889 in Linear Algebra for Puchuuu

Question

Let V be a vector space of 2×2 matrices over R. Show that the set S defined by S={(a,b)(c,d)belongs to V :a+b=0} is a subspace of R

Answer 1

• First of all, let us remark that the zero matrix satisfies the property $a+b=0$ so $0\in S$

Let $M, N$ be matrices in $S$ and $\lambda$ $\in \mathbb{R}$ a scalar. We will denote $a_M$ the top-left coefficient of a matrix $M$ and $b_M$ the top-right one (as in the question).

• By definition of a sum of two matrices, we have $a_{M+N}=a_M+a_N$, same for $b_{M+N}$, so we have $a_{M+N}+b_{M+N}=(a_M+b_M)+(a_N+b_N)=0$ and thus $M+N\in S$
• By definition of a scalar multiplication we have $a_{\lambda M} = \lambda a_M$, same for $b_{\lambda M}$ and thus we have $a_{\lambda M}+b_{\lambda M}= \lambda(a_M+b_M)=0$ and thus $\lambda M\in S$.

Therefore, $S$ is a vector subspace of $V$.